Reduced Basis Method for Finite Volume Approximations of Parametrized Evolution Equations
Reduced Basis Method for Finite Volume Approximations of Parametrized Evolution Equations
复制标题
参数化演化方程有限体积逼近的减基法
DOI：
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发表时间：
2006
期刊：
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通讯作者：
Mario Ohlberger
中科院分区：
文献类型：
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作者：
B. Haasdonk;Mario Ohlberger
通过提供近似求解过程和有效的误差估计，减基(RB)技术的模型降阶方法提供了对参数偏微分方程(PDE)的有效处理。到目前为止，RB方法主要用于椭圆型和抛物型问题的有限元格式。在当前的研究中，我们将该方法推广到一般的发展格式，如抛物型和双曲型发展方程的有限体积格式。新的理论贡献是提出了一般发展问题的减缩基近似格式，并在不同范数下得到了严格的先验误差估计。在算法上，实现了方案和误差估计器的离线/在线分解。这是在多个模拟请求的情况下快速在线计算的基础。我们介绍了一种新的基于后验误差估计器的离线基生成算法，该算法结合了现有方法的思想。实例对流扩散问题的数值实验证明了该方法的有效性。
The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (PDEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for general evolution problems and the derivation of rigorous aposteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized. This is the basis for a rapid online computation in case of multiple-simulation requests. We introduce a new offline basis-generation algorithm based on our a posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.